I am sorry to have to say this, because you have clearly put a lot of effort into your ideas. I have spent many hours trying to understand these, but I am no clearer than I was at the start.

Sorry.

Bob

Hi bob!

Here initial postulate of STRUCTURAL ANALYSIS:

I don't decide to translate it into English not to lose meaning.

Hi to All

The Fundamental Theorem of Calculus proves that:

1. That differentiation and antidifferentiation are essentially inverse processes:

2. How to evaluate a definite integral using the antiderivative of the integrand:

where F is an antiderivative of f (i.e., F' = f).

I think that on it the correct theory comes to an end.

Nobody proved it and it doesn't make sense.

Because:

]]>

I ask you about integrals, and you write me algebra.

When I am asked to explain something I go back to the basic theory.

What I have shown you **is an explanation of integration.**

I have offered to teach you calculus but you say you are already an expert. ?? You do not recognise what I have shown. ??

I think this is the end of this thread for me.

You misuse calculus but cannot understand my explanation of what is wrong.

I offer to help and you insist you need no help.

My opinion is that your theories will never be accepted by other mathematicians for these reasons:

(i) You say calculus has many errors, but all the examples you have given show only that you do not understand calculus.

(ii) Nevertheless, you use calculus to start a new theory.

(iii) Your explanations of the new theory are incomplete because you introduce functions, equations and graphs without explanation and that contradict known facts.

eg. Tangent lines have no direction.

eg. Opposite sides of a rectangle have different lengths.

eg. The radius and the height of a cone are not related.

I am sorry to have to say this, because you have clearly put a lot of effort into your ideas. I have spent many hours trying to understand these, but I am no clearer than I was at the start.

Sorry.

Bob

]]>I will make it as simple as I can.

I have a cone. The base radius is 4 cm. The height (from the vertex downward) is 8 cm.

I have a cylinder. The base radius is 4 cm. The height (from the top downward) is 8 cm.

I will calculate the volume of the cylinder in 4 slices.

I will calculate the volume of the **yellow** slices for the cone.

I will calculate the volume of the **green** slices for the cone.

Note:

For the cone

**I chose my values for R and H but they are related by a formula.**

Bob

]]>But I will teach you to use the rules properly.

works for both.

Bob

You didn't explain to me as it is possible using to receive ONE formula TWO various answers one of which is 3 times more than another.

This is THREE volumes !!!

So in Structural Analysis.

Write as it looks at you:

]]>

You could teach me to that how to define the rule for integration.

Not to define the rules. Other before me have already done that.

But I will teach you to use the rules properly.

For both the cylinder and the cone, the slice is (pi r^2) and the thickness or width is 'dh'.

So

works for both.

But the difference is that a cylinder has the same radius throughout its length => pi r^2 is constant,

whereas for the cone the radius changes along the axis.** pi r^2 is not constant.** It varies with h

So when you integrate you must change the 'r' into a function of 'h'.

If you do not do this you calculate the volume of a cylinder not a cone.

Bob

]]>21122012 wrote:You here understand everything?

No.

Bob

Hi bob

General view of these two expressions the identical:

You could teach me to that how to define the rule for integration. From where it is known that in one case one of variables for other variable is a constant and in other case they depend from each other. How to you it is prompted by a formula? Or each person establishes calculation rules itself voluntarily. After all answers turn out different.

]]>You here understand everything?

No. We still have not resolved post 22

Bob

]]>Hi Bob

I think I finally understand what he means by schedule- graph!

Yes but it not I am the robot so translates.

Here so it translated your phrase:

On English: "I think I finally understand what he means by schedule- grap"

->

On Russian: "Я думаю, что наконец понимаю то, что он подразумевает графиком - граф!"

]]>I think I finally understand what he means by schedule- graph!

]]>You here understand everything?

from left to right:

1. The segment of line.

2. The rectangular system of coordinates (for drawing of charts and schedules of independent sizes).

3. Cartesian coordinate system (for schedules of dependent sizes)

4.Cartesian coordinate system

Write please the total and partial derivatives for these two cases.

I do not understand you.

total derivative. What do you want me to differentiate with respect to?

1.

u = pi r^2 ≠ f(h)

OK?

2. Say u = pi r^2 = pi (kh)^2

What now?

Bob

]]>...

...

Similarly, the total derivative with respect to h is:

...

Bob

Wikipedia:

Identical type of expressions - decide differently. On the ode of a variable the derivative undertakes, other letter registers.

bob bundy wrote:

2122012 wrote:.I do not understand you.

total derivative. What do you want me to differentiate with respect to?

Two functions, each function of two variables, in one option dependent are given, in the other - of the independent - is unclear on what function to consider a total derivative...

This is cardsharpering instead of exact science.

]]>Look this. You equate two red areas to which shooters point. These areas aren't equal.

I think I have worked out why you keep getting this wrong.

It is true: integration can work out areas

Definition:

**Integration is a summation process.**

dictionary.com = "ʃ the limit of an increasingly large number of increasingly smaller quantities"

You can add areas to get volumes, or velocities to get distance, or moments about a point to get the total moment.

I am **not** integrating to get the area under a graph.

I am working out the volume of a small slice of the shape with 'dh' representing the variable I am integrating with respect to.

To do this the function to be integrated** must be a function of 'h'.**

And you either put in upper and lower limits for h, or put in a constant of integration.

Only when you understand and accept this, will you be able to do integration properly.

**Note: The integration symbol ∫ is a stylised letter 's' to stand for 'sum'**

Now you seem to have a continuing problem with r = kh.

'r' and 'h' are variables. As you add up successive volumes, r may vary. If it does then you must write r in terms of h.

You referred to wikipedia. Here is the **full** text from the bit you quoted:

If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio k,

This gives the total derivative with respect to r:

Which simplifies to:Similarly, the total derivative with respect to h is:

Note: I used r = kh. Wiki used h = kr. So our values of k are reciprocals of each other. Of course, this doesn't alter my calculus.

The picture below is from Wolfram Alpha.** It shows that, for a cone, r and h are related.**

Bob

]]>hi 21122012,

Oh thank you. I am so pleased you have decided to ask me to explain integration.

To work out a volume you divide the solid into thin slices, each one dh in thickness and add them up

For a cylinder each slice is a circle with a radius of r.

Now add them up

Now, and this is the important bit, for a cylinder, every slice is the

same size, so the pi r^2 term isconstantas h varies.

Everything is up to this point all is correct

bob bundy wrote:

If V = 0 when h = 0 then C = 0

I don't understand this thought!

bob bundy wrote:

For a cone each slice is again a circle with a radius of r.

But the circles are not all the same size. As h increases from o to H, the radius changes from 0 to R.

So \pi r^2 is

not constant.

Gallantly!

bob bundy wrote:

STOP!!!

Here mistake!!! I constantly speak to you about it, but you don't hear me!!!

It not algebra! At the left you have two independent variables therefore the result will be one. On the right two dependent variables therefore the result will be another. These two expressions aren't EQUAL! Use WolframAlfa, it will yield to you two various results!

Look this. You equate two red areas to which shooters point. These areas aren't equal.

bob bundy wrote:

So the result is different.

Do you understand now how integration works ?

Add up the slices but take account of whether they are all the same size, or change as h changes.

Bob

Bob you don't make laugh me. I and WolframAlpha we know as integration works.

P.S.

bob bundy wrote:

hi 21122012,

...

...

Bob

BECAUSE:

bob bundy wrote:

Do you understand now how integration works ?

Bob

P.P.S.

Bob!

It already amused me. Let's talk about the serious. Here one from the most important keys to Structural Analysis. You here understand everything?

from left to right:

1. The segment of line.

2. The rectangular system of coordinates (for drawing of charts and schedules of independent sizes).

3. Cartesian coordinate system (for schedules of dependent sizes)

4.Cartesian coordinate system